Introduction to Generalized Linear Models Prepared by Louise Francis Francis Analytics and Actuarial...

Introduction to Generalized Linear Models

Prepared byLouise Francis Francis Analytics and Actuarial Data Mining, Inc.October 3, 2004

ObjectivesGentle introduction to Linear Models and Generalized Linear ModelsIllustrate some simple applicationsShow examples in commonly available softwareWhich model(s) to use?Practical issues

A Brief Introduction to Regression

One of most common statistical methods fits a line to dataModel: Y = a+bx + error Error assumed to be Normal

Chart1

199119911991199119915410.36894801813715.4316490031

199219921992199219924868.22401916743985.4514762824

199319931993199319934392.97032555464255.4713035617

199419941994199419944190.79939668184525.4911308411

199519951995199519953892.14956055424795.5109581204

199619961996199619963493.68593076815065.5307853997

199719971997199719974529.44911890785335.5506126791

199819981998199819984977.45413790195605.5704399584

199919991999199919995452.77807391365875.5902672377

200020002000200020005726.90586661316145.6100945171

200120012001200120016686.81635002886415.6299217964

200220022002200220026885.12381307026685.6497490757

200320032003200320038855.43242364836955.6695763551

Severity

Fitted Y

Year

Severity

Workers Comp Sevirity Trend

wORKERS cOMP

Paid LossesInc LossesIBNR and BulkIncurredPaid to IncurredCase

Policy Year12345678910Policy Year12345678910Policy Year12345678910Policy Year12345678910Accident Year12345678910Accident Year

PriorPriorPriorPrior0.00.00.0PriorPrior

19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991

199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992

199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993

199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994

199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995

199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)

19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)

19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)

19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)

20004,534,0352002200220000.020000.02000(4,534,035)

246195205056278

$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)

Age to Age FactorsAge to Age Factors

Paid Losses$39,739Incurred(75,792,474)

Ratio IBNR to CASE(0.00)

Policy Year12345678910Policy Year12345678910Ratio Case to Incurred(570.10)

PriorPriorRatio Case to Incurred(534.82)(536.08)(616.98)

19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012

19921.0221.02119921.0171.013

19931.0311.03019931.0171.019

19941.0451.035Claims19941.0281.022

19951.0731.054Alt19951.0421.039

19961.1371.085Claim DFReported ClmsW PaymentCNPClaimsUlt ClaimsUlt LossesUlt SeveritySeverity on CWPRpt Inc LossesRpt SevLDFSelectedClose RateExtra DevTrended SevSelectedPaidInc to Pd19961.0691.047

19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076

19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134

19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450

20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000

Bests DevRemainder

Selected2.251.31.141.081.051.031.0251.021.0131.11.08293550031.134.176534601Average1.4131.1231.0641.0371.0311.0161.0151.0121.012

Cumulative4.5372.0171.5511.3611.2601.2001.1651.1371.1141.100Open ClaimsCase ReservesRes/Oprn1.12316149161.1537341867Selected1.421.131.061.041.031.021.021.0151.011.07

3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070

1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111

2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700

20001,386,125(4,534,035)(3,271.01)

4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006

WC

Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeverity

Policy Year

Prior

19913944772913017,291.00$5,410.37

19923251066783416,678.00$4,868.22

19933049669422816,942.00$4,392.974375.3228120516

19942778566303316,630.00$4,190.804170.0435239382

19952612867134416,713.00$3,892.153866.8047950274

19962351667315716,731.00$3,493.693464.3488509134

19972853162997416,299.00$4,529.454476.8554840734

19983204764321151.0016,438.43$4,977.454894.9137009317

19993467863471831.0026,359.69$5,452.785310.5666156202

20003677063573391.016,420.57$5,726.91365491.3381123058

20013949157906161.025,905.80$6,686.82246164.6893537309

20023882953715211.055,639.55$6,885.12246678.1688919716

200342232414711671.154,769.05$8,855.4312

197015.7

197110.9

197211.1

19739.6

197415.7

197514.2

197620.4

197716.1

197810.1

197916.5

198015.2

198115.5

198213.2

19838

19848.2

198510

19865

198710.9

198815

198912.9

199012.9

199110.9

19928

19937.3

19941.8

19953.1

19961.9

19973.6

19984

19993

20007

20019.4

20029.7

Sheet1

WC

Losses & LAE IncurredRpt ClaimsOS ClaimsFactorClaimsSeveritySUMMARY OUTPUT

Accident Year

PriortRegression Statistics

(1)(2)(3)

Multiple R0.7219364186

YearSeverityFitted YXR Square0.5211921926

19913944772913017,291.00$5,410$3,7151Adjusted R Square0.4776642101

19923251066783416,678.00$4,868$3,9852Standard Error1052.729195738

19933049669422816,942.00$4,393$4,2553Observations13

19942778566303316,630.00$4,191$4,5254

19952612867134416,713.00$3,892$4,7965ANOVA

19962351667315716,731.00$3,494$5,0666dfSSMSFSignificance F

19972853162997416,299.00$4,529$5,3367Regression113269748.696560913269748.696560911.97372730570.0053302696

19983204764321151.0016,438.43$4,977$5,6068Residual1112190626.35514981108238.75955907

19993467863471831.0026,359.69$5,453$5,8769Total1225460375.0517107

20003677063573391.016,420.57$5,727$6,14610

20013949157906161.025,905.80$6,687$6,41611CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

20023882953715211.055,639.55$6,885$6,68612Intercept3445.4118217237619.37154364585.56275446790.00016948892082.18355612554808.6400873222082.18355612554808.640087322

200342232414711671.154,769.05$8,855$6,95613X Variable 1270.019827279378.0334796973.46030740050.005330269698.2692096133441.770444945498.2692096133441.7704449454

RESIDUAL OUTPUTPROBABILITY OUTPUT

ObservationPredicted YResidualsPercentileY

13715.43164900311694.9372990153.84615384623493.6859307681

23985.4514762824882.77254288511.53846153853892.1495605542

34255.4713035617137.499021992919.23076923084190.7993966818

44525.4911308411-334.691734159326.92307692314392.9703255546

54795.5109581204-903.361397566234.61538461544529.4491189078

65065.5307853997-1571.844854631642.30769230774868.2240191674

75335.5506126791-806.1014937713504977.4541379019

85605.5704399584-628.116302056557.69230769235410.3689480181

95875.5902672377-422.812193324165.38461538465452.7780739136

106145.6100945171-418.70422790473.07692307695726.9058666131

116415.6299217964271.186428232480.76923076926686.8163500288

126685.6497490757199.474063994588.46153846156885.1238130702

136955.66957635511899.762847293396.15384615388855.4324236483

Sheet1

1694.937299015

882.772542885

137.4990219929

-334.6917341593

-903.3613975662

-1571.8448546316

-806.1014937713

-628.1163020565

-422.8121933241

-418.704227904

271.1864282324

199.4740639945

1899.7628472933

X Variable 1

Residuals

X Variable 1 Residual Plot

Sheet2

3715.43164900313715.4316490031

3985.45147628243985.4514762824

4255.47130356174255.4713035617

4525.49113084114525.4911308411

4795.51095812044795.5109581204

5065.53078539975065.5307853997

5335.55061267915335.5506126791

5605.57043995845605.5704399584

5875.59026723775875.5902672377

6145.61009451716145.6100945171

6415.62992179646415.6299217964

6685.64974907576685.6497490757

6955.66957635516955.6695763551

Y

Predicted Y

Year Index

Y

Workers Compensation Claim Severity

Sheet3

3493.6859307681

3892.1495605542

4190.7993966818

4392.9703255546

4529.4491189078

4868.2240191674

4977.4541379019

5410.3689480181

5452.7780739136

5726.9058666131

6686.8163500288

6885.1238130702

8855.4324236483

Sample Percentile

Y

Normal Probability Plot

Severity

Fitted Y

Year

Severity


A Brief Introduction to Regression

Fits line that minimizes squared deviation between actual and fitted values

Simple Formula for Fitting Line

wORKERS cOMP




19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991

199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992

199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993

199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994

199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995

199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)

19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)

19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)

19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)

20004,534,0352002200220000.020000.02000(4,534,035)

246195205056278

$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)






19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012

19921.0221.02119921.0171.013

19931.0311.03019931.0171.019

19941.0451.035Claims19941.0281.022

19951.0731.054Alt19951.0421.039


19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076

19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134

19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450

20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000

Bests DevRemainder



3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070

1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111

2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700

20001,386,125(4,534,035)(3,271.01)

4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006

WC


Policy Year

Prior

19913944772913017,291.00$5,410.37

19923251066783416,678.00$4,868.22

19933049669422816,942.00$4,392.974375.3228120516

19942778566303316,630.00$4,190.804170.0435239382

19952612867134416,713.00$3,892.153866.8047950274

19962351667315716,731.00$3,493.693464.3488509134

19972853162997416,299.00$4,529.454476.8554840734

19983204764321151.0016,438.43$4,977.454894.9137009317

19993467863471831.0026,359.69$5,452.785310.5666156202

20003677063573391.016,420.57$5,726.91365491.3381123058

20013949157906161.025,905.80$6,686.82246164.6893537309

20023882953715211.055,639.55$6,885.12246678.1688919716

200342232414711671.154,769.05$8,855.4312

197015.7

197110.9

197211.1

19739.6

197415.7

197514.2

197620.4

197716.1

197810.1

197916.5

198015.2

198115.5

198213.2

19838

19848.2

198510

19865

198710.9

198815

198912.9

199012.9

199110.9

19928

19937.3

19941.8

19953.1

19961.9

19973.6

19984

19993

20007

20019.4

20029.7

Sheet1

WC

SUMMARY OUTPUT

Regression Statistics

(1)(2)(3)(4)(5)(6)(7)(8)

AccidentSeverityMultiple R0.7219364186

YearYFitted YXR Square0.5211921926

19913944772913017,291.00$5,410$3,7151$75-6(448.91)36Adjusted R Square0.4776642101

19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.729195738

19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13

19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259

19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA

19962351667315716,731.00$3,494$5,0666$(1,842)-11,841.861dfSSMSFSignificance F

19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.696560913269748.696560911.97372730570.0053302696

19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.35514981108238.75955907

19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.0517107

20003677063573391.016,420.57$5,727$6,14610$39131,174.079

20013949157906161.025,905.80$6,687$6,41611$1,35145,405.0616CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.4118217237619.37154364585.56275446790.00016948892082.18355612554808.6400873222082.18355612554808.640087322

200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.019827279378.0334796973.46030740050.005330269698.2692096133441.770444945498.2692096133441.7704449454

Sum/Average$5,336749,143.61182.00

B270.02

3,445.41



13715.43164900311694.9372990153.84615384623493.6859307681

23985.4514762824882.77254288511.53846153853892.1495605542

34255.4713035617137.499021992919.23076923084190.7993966818

44525.4911308411-334.691734159326.92307692314392.9703255546

54795.5109581204-903.361397566234.61538461544529.4491189078

65065.5307853997-1571.844854631642.30769230774868.2240191674

75335.5506126791-806.1014937713504977.4541379019

85605.5704399584-628.116302056557.69230769235410.3689480181

95875.5902672377-422.812193324165.38461538465452.7780739136

106145.6100945171-418.70422790473.07692307695726.9058666131

116415.6299217964271.186428232480.76923076926686.8163500288

126685.6497490757199.474063994588.46153846156885.1238130702

136955.66957635511899.762847293396.15384615388855.4324236483

Sheet1

1694.937299015

882.772542885

137.4990219929

-334.6917341593

-903.3613975662

-1571.8448546316

-806.1014937713

-628.1163020565

-422.8121933241

-418.704227904

271.1864282324

199.4740639945

1899.7628472933

X Variable 1

Residuals


Sheet2

3493.6859307681

3892.1495605542

4190.7993966818

4392.9703255546

4529.4491189078

4868.2240191674

4977.4541379019

5410.3689480181

5452.7780739136

5726.9058666131

6686.8163500288

6885.1238130702

8855.4324236483

Sample Percentile

Y


Sheet3

199119911991199119915410.36894801813715.4316490031

199219921992199219924868.22401916743985.4514762824

199319931993199319934392.97032555464255.4713035617

199419941994199419944190.79939668184525.4911308411

199519951995199519953892.14956055424795.5109581204

199619961996199619963493.68593076815065.5307853997

199719971997199719974529.44911890785335.5506126791

199819981998199819984977.45413790195605.5704399584

199919991999199919995452.77807391365875.5902672377

200020002000200020005726.90586661316145.6100945171

200120012001200120016686.81635002886415.6299217964

200220022002200220026885.12381307026685.6497490757

200320032003200320038855.43242364836955.6695763551

Y

Fitted Y

Year

Severity


MBD001271B2.unknown

MBD0012AE08.unknown

MBD0012EFC8.unknown

MBD00129396.unknown

MBD00126489.unknown

Excel Does RegressionInstall Data Analysis Tool Pak (Add In) that comes with ExcelClick Tools, Data Analysis, Regression

Goodness of Fit StatisticsR2: (SS Regression/SS Total)percentage of variance explainedF statistic: (MS Regression/MS Resid)significance of regressionT statistics: Uses SE of coefficient to determine if it is significant significance of coefficientsIt is customary to drop variable if coefficient not significant

Note SS = Sum squared of errors

Output of Excel Regression Procedure

wORKERS cOMP




19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991

199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992

199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993

199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994

199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995

199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)

19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)

19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)

19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)

20004,534,0352002200220000.020000.02000(4,534,035)

246195205056278

$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)






19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012

19921.0221.02119921.0171.013

19931.0311.03019931.0171.019

19941.0451.035Claims19941.0281.022

19951.0731.054Alt19951.0421.039


19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076

19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134

19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450

20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000

Bests DevRemainder



3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070

1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111

2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700

20001,386,125(4,534,035)(3,271.01)

4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006

WC


Policy Year

Prior

19913944772913017,291.00$5,410.37

19923251066783416,678.00$4,868.22

19933049669422816,942.00$4,392.974375.3228120516

19942778566303316,630.00$4,190.804170.0435239382

19952612867134416,713.00$3,892.153866.8047950274

19962351667315716,731.00$3,493.693464.3488509134

19972853162997416,299.00$4,529.454476.8554840734

19983204764321151.0016,438.43$4,977.454894.9137009317

19993467863471831.0026,359.69$5,452.785310.5666156202

20003677063573391.016,420.57$5,726.91365491.3381123058

20013949157906161.025,905.80$6,686.82246164.6893537309

20023882953715211.055,639.55$6,885.12246678.1688919716

200342232414711671.154,769.05$8,855.4312

197015.7

197110.9

197211.1

19739.6

197415.7

197514.2

197620.4

197716.1

197810.1

197916.5

198015.2

198115.5

198213.2

19838

19848.2

198510

19865

198710.9

198815

198912.9

199012.9

199110.9

19928

19937.3

19941.8

19953.1

19961.9

19973.6

19984

19993

20007

20019.4

20029.7

Sheet1

WC

SUMMARY OUTPUT

tRegression Statistics

(1)(2)(3)(4)(5)(6)(7)(8)




19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73

19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00

19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259

19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA


19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01

19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76

19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05

20003677063573391.016,420.57$5,727$6,14610$39131,174.079


20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64

200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77

Sum/Average$5,336749,143.61182.00

B270.02

3,445.41



13715.43164900311694.9372990153.84615384623493.6859307681

23985.4514762824882.77254288511.53846153853892.1495605542

34255.4713035617137.499021992919.23076923084190.7993966818

44525.4911308411-334.691734159326.92307692314392.9703255546

54795.5109581204-903.361397566234.61538461544529.4491189078

65065.5307853997-1571.844854631642.30769230774868.2240191674

75335.5506126791-806.1014937713504977.4541379019

85605.5704399584-628.116302056557.69230769235410.3689480181

95875.5902672377-422.812193324165.38461538465452.7780739136

106145.6100945171-418.70422790473.07692307695726.9058666131

116415.6299217964271.186428232480.76923076926686.8163500288

126685.6497490757199.474063994588.46153846156885.1238130702

136955.66957635511899.762847293396.15384615388855.4324236483

Sheet1

X Variable 1

Residuals


Sheet2

Sample Percentile

Y


Sheet3

Y

Fitted Y

Year

Severity


MBD001271B2.unknown

MBD0012AE08.unknown

MBD0012EFC8.unknown

MBD00129396.unknown

MBD00126489.unknown

Assumptions of RegressionErrors independent of value of XErrors independent of value of YErrors independent of prior errorsErrors are from normal distributionWe can test these assumptions

Other Diagnostics: Residual PlotPoints should scatter randomly around zeroIf not, a straight line probably is not be appropriate

Chart1

1694.937299015

882.772542885

137.4990219929

-334.6917341593

-903.3613975662

-1571.8448546316

-806.1014937713

-628.1163020565

-422.8121933241

-418.704227904

271.1864282324

199.4740639945

1899.7628472933

X Variable 1

Residuals

wORKERS cOMP




19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991

199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992

199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993

199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994

199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995

199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)

19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)

19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)

19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)

20004,534,0352002200220000.020000.02000(4,534,035)

246195205056278

$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)






19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012

19921.0221.02119921.0171.013

19931.0311.03019931.0171.019

19941.0451.035Claims19941.0281.022

19951.0731.054Alt19951.0421.039


19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076

19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134

19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450

20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000

Bests DevRemainder



3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070

1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111

2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700

20001,386,125(4,534,035)(3,271.01)

4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006

WC


Policy Year

Prior

19913944772913017,291.00$5,410.37

19923251066783416,678.00$4,868.22

19933049669422816,942.00$4,392.974375.3228120516

19942778566303316,630.00$4,190.804170.0435239382

19952612867134416,713.00$3,892.153866.8047950274

19962351667315716,731.00$3,493.693464.3488509134

19972853162997416,299.00$4,529.454476.8554840734

19983204764321151.0016,438.43$4,977.454894.9137009317

19993467863471831.0026,359.69$5,452.785310.5666156202

20003677063573391.016,420.57$5,726.91365491.3381123058

20013949157906161.025,905.80$6,686.82246164.6893537309

20023882953715211.055,639.55$6,885.12246678.1688919716

200342232414711671.154,769.05$8,855.4312

197015.7

197110.9

197211.1

19739.6

197415.7

197514.2

197620.4

197716.1

197810.1

197916.5

198015.2

198115.5

198213.2

19838

19848.2

198510

19865

198710.9

198815

198912.9

199012.9

199110.9

19928

19937.3

19941.8

19953.1

19961.9

19973.6

19984

19993

20007

20019.4

20029.7

Sheet1

WC

SUMMARY OUTPUT


(1)(2)(3)(4)(5)(6)(7)(8)




19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73

19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00

19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259

19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA


19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01

19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76

19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05

20003677063573391.016,420.57$5,727$6,14610$39131,174.079


20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64

200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77

Sum/Average$5,336749,143.61182.00

B270.02

3,445.41



13715.43164900311694.9372990153.84615384623493.6859307681

23985.4514762824882.77254288511.53846153853892.1495605542

34255.4713035617137.499021992919.23076923084190.7993966818

44525.4911308411-334.691734159326.92307692314392.9703255546

54795.5109581204-903.361397566234.61538461544529.4491189078

65065.5307853997-1571.844854631642.30769230774868.2240191674

75335.5506126791-806.1014937713504977.4541379019

85605.5704399584-628.116302056557.69230769235410.3689480181

95875.5902672377-422.812193324165.38461538465452.7780739136

106145.6100945171-418.70422790473.07692307695726.9058666131

116415.6299217964271.186428232480.76923076926686.8163500288

126685.6497490757199.474063994588.46153846156885.1238130702

136955.66957635511899.762847293396.15384615388855.4324236483

Sheet1

X Variable 1

Residuals

Sheet2

Sample Percentile

Y


Sheet3

Y

Fitted Y

Year

Severity


MBD001271B2.unknown

MBD0012AE08.unknown

MBD0012EFC8.unknown

MBD00129396.unknown

MBD00126489.unknown

Other Diagnostics: Normal PlotPlot should be a straight lineOtherwise residuals not from normal distribution

Chart2

3493.6859307681

3892.1495605542

4190.7993966818

4392.9703255546

4529.4491189078

4868.2240191674

4977.4541379019

5410.3689480181

5452.7780739136

5726.9058666131

6686.8163500288

6885.1238130702

8855.4324236483

Sample Percentile

Y


wORKERS cOMP




19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991

199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992

199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993

199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994

199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995

199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)

19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)

19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)

19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)

20004,534,0352002200220000.020000.02000(4,534,035)

246195205056278

$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)






19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012

19921.0221.02119921.0171.013

19931.0311.03019931.0171.019

19941.0451.035Claims19941.0281.022

19951.0731.054Alt19951.0421.039


19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076

19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134

19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450

20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000

Bests DevRemainder



3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070

1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111

2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700

20001,386,125(4,534,035)(3,271.01)

4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006

WC


Policy Year

Prior

19913944772913017,291.00$5,410.37

19923251066783416,678.00$4,868.22

19933049669422816,942.00$4,392.974375.3228120516

19942778566303316,630.00$4,190.804170.0435239382

19952612867134416,713.00$3,892.153866.8047950274

19962351667315716,731.00$3,493.693464.3488509134

19972853162997416,299.00$4,529.454476.8554840734

19983204764321151.0016,438.43$4,977.454894.9137009317

19993467863471831.0026,359.69$5,452.785310.5666156202

20003677063573391.016,420.57$5,726.91365491.3381123058

20013949157906161.025,905.80$6,686.82246164.6893537309

20023882953715211.055,639.55$6,885.12246678.1688919716

200342232414711671.154,769.05$8,855.4312

197015.7

197110.9

197211.1

19739.6

197415.7

197514.2

197620.4

197716.1

197810.1

197916.5

198015.2

198115.5

198213.2

19838

19848.2

198510

19865

198710.9

198815

198912.9

199012.9

199110.9

19928

19937.3

19941.8

19953.1

19961.9

19973.6

19984

19993

20007

20019.4

20029.7

Sheet1

WC

SUMMARY OUTPUT


(1)(2)(3)(4)(5)(6)(7)(8)




19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73

19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00

19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259

19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA


19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01

19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76

19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05

20003677063573391.016,420.57$5,727$6,14610$39131,174.079


20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64

200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77

Sum/Average$5,336749,143.61182.00

B270.02

3,445.41



13715.43164900311694.9372990153.84615384623493.6859307681

23985.4514762824882.77254288511.53846153853892.1495605542

34255.4713035617137.499021992919.23076923084190.7993966818

44525.4911308411-334.691734159326.92307692314392.9703255546

54795.5109581204-903.361397566234.61538461544529.4491189078

65065.5307853997-1571.844854631642.30769230774868.2240191674

75335.5506126791-806.1014937713504977.4541379019

85605.5704399584-628.116302056557.69230769235410.3689480181

95875.5902672377-422.812193324165.38461538465452.7780739136

106145.6100945171-418.70422790473.07692307695726.9058666131

116415.6299217964271.186428232480.76923076926686.8163500288

126685.6497490757199.474063994588.46153846156885.1238130702

136955.66957635511899.762847293396.15384615388855.4324236483

Sheet1

X Variable 1

Residuals

Sheet2

Sample Percentile

Y


Sheet3

Y

Fitted Y

Year

Severity


MBD001271B2.unknown

MBD0012AE08.unknown

MBD0012EFC8.unknown

MBD00129396.unknown

MBD00126489.unknown

Test for autocorrelated errorsAutocorrelation often present in time series dataDurban Watson statistic:If residuals uncorrelated, this is near 2

Durban Watson StatisticIndicates autocorrelation present

wORKERS cOMP




19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991

199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992

199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993

199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994

199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995

199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)

19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)

19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)

19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)

20004,534,0352002200220000.020000.02000(4,534,035)

246195205056278

$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)






19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012

19921.0221.02119921.0171.013

19931.0311.03019931.0171.019

19941.0451.035Claims19941.0281.022

19951.0731.054Alt19951.0421.039


19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076

19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134

19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450

20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000

Bests DevRemainder



3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070

1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111

2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700

20001,386,125(4,534,035)(3,271.01)

4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006

WC


Policy Year

Prior

19913944772913017,291.00$5,410.37

19923251066783416,678.00$4,868.22

19933049669422816,942.00$4,392.974375.3228120516

19942778566303316,630.00$4,190.804170.0435239382

19952612867134416,713.00$3,892.153866.8047950274

19962351667315716,731.00$3,493.693464.3488509134

19972853162997416,299.00$4,529.454476.8554840734

19983204764321151.0016,438.43$4,977.454894.9137009317

19993467863471831.0026,359.69$5,452.785310.5666156202

20003677063573391.016,420.57$5,726.91365491.3381123058

20013949157906161.025,905.80$6,686.82246164.6893537309

20023882953715211.055,639.55$6,885.12246678.1688919716

200342232414711671.154,769.05$8,855.4312

197015.71.1571

197110.91.1091.109

197211.11.1111.232099

19739.61.0961.350380504

197415.71.1571.5623902431

197514.21.1421.7842496577

197620.41.2042.1482365878

197716.11.1612.4941026785

197810.11.1012.746007049

197916.51.1653.1990982121

198015.21.1523.6853611403

198115.51.1554.256592117

198213.21.1324.8184622765

198381.085.2039392586

19848.21.0825.6306622778

1985101.16.1937285056

198651.056.5034149309

198710.91.1097.2122871583

1988151.158.2941302321

198912.91.1299.364073032

199012.91.12910.5720384531

199110.91.10911.7243906445

199281.0812.6623418961

19937.31.07313.5866928545

19941.81.01813.8312533259

19953.11.03114.260022179

19961.91.01914.5309626004

19973.61.03615.054077254

199841.0415.6562403442

199931.0316.1259275545

200071.0717.2547424833

20019.41.09418.8766882768

20029.71.09720.7077270396

2003151.1523.8138860955

Line Fit

WC

SUMMARY OUTPUT


(1)(2)(3)(4)(5)(6)(7)(8)




19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73

19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00

19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259

19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA


19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01

19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76

19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05

20003677063573391.016,420.57$5,727$6,14610$39131,174.079


20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64

200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77

Sum/Average$5,336749,143.61182.00

B270.02

3,445.41



13715.43164900311694.9372990153.84615384623493.6859307681

23985.4514762824882.77254288511.53846153853892.1495605542

34255.4713035617137.499021992919.23076923084190.7993966818

44525.4911308411-334.691734159326.92307692314392.9703255546

54795.5109581204-903.361397566234.61538461544529.4491189078

65065.5307853997-1571.844854631642.30769230774868.2240191674

75335.5506126791-806.1014937713504977.4541379019

85605.5704399584-628.116302056557.69230769235410.3689480181

95875.5902672377-422.812193324165.38461538465452.7780739136

106145.6100945171-418.70422790473.07692307695726.9058666131

116415.6299217964271.186428232480.76923076926686.8163500288

126685.6497490757199.474063994588.46153846156885.1238130702

136955.66957635511899.762847293396.15384615388855.4324236483

Line Fit

0

0

0

0

0

0

0

0

0

0

0

0

0

X Variable 1

Residuals

Expo Fit

0

0

0

0

0

0

0

0

0

0

0

0

0

Sample Percentile

Y


Econo Model

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

Y

Fitted Y

Year

Severity

Workers Comp Severity Trend

Econo Model (2)

WC

SUMMARY OUTPUT


(1)(2)(3)(4)(5)(5)(6)(7)(8)Multiple R0.6992898719

AccidentSeverityln(Severity)R Square0.489006325

YearYYFitted SevXAdjusted R Square0.4425523545

YearSeverityFittedStandard Error0.1904278372

19913944772913017,291.00$5,4108.60$3,9321$75-6(448.91)361991$5,410$3,932Observations13

19923251066783416,678.00$4,8688.49$4,1162$(467)-52,336.63251992$4,868$4,116

19933049669422816,942.00$4,3938.39$4,3093$(943)-43,770.32161993$4,393$4,309ANOVA

19942778566303316,630.00$4,1918.34$4,5114$(1,145)-33,434.2591994$4,191$4,511dfSSMSFSignificance F

19952612867134416,713.00$3,8928.27$4,7225$(1,443)-22,886.8041995$3,892$4,722Regression10.38172667260.381726672610.52668523720.0078114247

19962351667315716,731.00$3,4948.16$4,9436$(1,842)-11,841.8611996$3,494$4,943Residual110.39889037280.0362627612

19972853162997416,299.00$4,5298.42$5,1757$(806)00.001997$4,529$5,175Total120.7806170454

19983204764321151.0016,438.43$4,9778.51$5,4178$(358)1(358.10)11998$4,977$5,417

19993467863471831.0026,359.69$5,4538.60$5,6719$1172234.4541999$5,453$5,671CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

20003677063573391.016,420.57$5,7278.65$5,93710$39131,174.0792000$5,727$5,937Intercept8.23034398770.112037914373.460346314907.98375007628.47693789937.98375007628.4769378993

20013949157906161.025,905.80$6,6878.81$6,21511$1,35145,405.06162001$6,687$6,215X Variable 10.04579737220.01411545043.24448535780.00781142470.01472945960.07686528480.01472945960.0768652848

20023882953715211.055,639.55$6,8858.84$6,50712$1,55057,747.87252002$6,885$6,507

200342232414711671.154,769.05$8,8559.09$6,81113$3,520621,119.29362003$8,855$6,811

Sum/Average$5,3368.58$5,175749,143.61182.00

B270.02RESIDUAL OUTPUTPROBABILITY OUTPUT

3,445.41

ObservationPredicted YResidualsStandard ResidualsPercentileY

18.27614135990.3199312071.75477100673.84615384628.1587125985

28.32193873210.16854573970.924446164611.53846153858.2667168703

38.36773610430.02002478470.109832710419.23076923088.3406467815

48.4135334765-0.072886695-0.39977175226.92307692318.387760889

58.4593308487-0.1926139784-1.056456567734.61538461548.4183556038

68.5051282209-0.3464156224-1.900033748242.30769230778.4904844718

78.5509255931-0.1325699893-0.7271249834508.512673822

88.5967229653-0.0840491433-0.460995978357.69230769238.5960725669

98.6425203375-0.0386398414-0.211933290265.38461538468.6038804961

108.6883177097-0.0353870342-0.194092168173.07692307698.6529306755

118.73411508190.07377807610.404660833280.76923076928.8078931579

128.77991245410.05720593990.313765342788.46153846158.837118394

138.82570982630.26307655661.442932430296.15384615389.0887863829

Econo Model (2)

0.0200247847

-0.072886695

-0.1926139784

-0.3464156224

-0.1325699893

-0.0840491433

-0.0386398414

-0.0353870342

0.0737780761

0.0572059399

0.2630765566

12

13

X Variable 1

Residuals

Econo Model Test

19.2307692308

26.9230769231

34.6153846154

42.3076923077

50

57.6923076923

65.3846153846

73.0769230769

80.7692307692

88.4615384615

96.1538461538

Sample Percentile

Y


Sheet3

199119911991199119915410.36894801818.3677361043

199219921992199219924868.22401916748.4135334765

199319931993199319934392.97032555468.4593308487

199419941994199419944190.79939668188.5051282209

199519951995199519953892.14956055428.5509255931

199619961996199619963493.68593076818.5967229653

199719971997199719974529.44911890788.6425203375

199819981998199819984977.45413790198.6883177097

199919991999199919995452.77807391368.7341150819

200020002000200020005726.90586661318.7799124541

200120012001200120016686.81635002888.8257098263

200220022002200220026885.12381307020

200320032003200320038855.43242364830

Y

Fitted Sev

Year

Severity


0.319931207

0.1685457397

0.0200247847

-0.072886695

-0.1926139784

-0.3464156224

-0.1325699893

-0.0840491433

-0.0386398414

-0.0353870342

0.0737780761

0.0572059399

0.2630765566

X Variable 1

Residuals


8.59607256698.2761413599

8.49048447188.3219387321

8.3877608898.3677361043

8.34064678158.4135334765

8.26671687038.4593308487

8.15871259858.5051282209

8.41835560388.5509255931

8.5126738228.5967229653

8.60388049618.6425203375

8.65293067558.6883177097

8.80789315798.7341150819

8.8371183948.7799124541

9.08878638298.8257098263

Y

Predicted Y

X Variable 1

Y

X Variable 1 Line Fit Plot

8.1587125985

8.2667168703

8.3406467815

8.387760889

8.4183556038

8.4904844718

8.512673822

8.5960725669

8.6038804961

8.6529306755

8.8078931579

8.837118394

9.0887863829

Sample Percentile

Y


0

0

0

0

0

0

0

0

0

0

0

0

0

Residuals

Predicted Y

Residual

Plot of Residuals

00

00

00

00

00

00

00

00

00

00

00

00

00

Severity

Fitted

Year

Severity

Exponential Trend

(1)(2)(3)(4)(5)(6)

WCHealth

YearSeverityIns IndexCPIFittedStd ResidualSUMMARY OUTPUT

19915,41011.7136.24,158.62.0

19924,86812.7140.34,496.60.6Regression Statistics

19934,39313.6144.54,818.6(0.7)Multiple R0.8988759252

19944,19113.8148.34,718.3(0.8)R Square0.8079779288

19953,89214.3152.44,716.4(1.3)Adjusted R Square0.7695735146

19963,49414.5156.94,581.0(1.7)Standard Error699.2105512592

19974,52915.1160.64,670.2(0.2)Observations13

19984,97715.7163.04,900.00.1

19995,45316.1166.64,960.60.8ANOVA

20005,72717.3172.25,320.60.6dfSSMSFSignificance F

20016,68718.9177.16,062.61.0Regression2.0020,571,421.1010285710.550894121.03867341820.0002610693

20026,88520.7179.97,084.8(0.3)Residual10.004,888,953.95488895.394992245

20038,85523.8184.08,873.9(0.0)Total12.0025,460,375.05

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept5,903.563,623.591.60.1-2170.313977.4-2170.313977.4

X Variable 1667.63172.153.90.0284.11051.2284.11051.2

Ln(severity)Ln(HI)Ln(CPI)X Variable 2-70.2837.96-1.90.1-154.914.3-154.914.3

8.59607256692.46167134244.9141243937

8.49048447182.53863238354.9437829871

8.3877608892.60909084724.9732795076

8.34064678152.62693076534.9989000379RESIDUAL OUTPUTPROBABILITY OUTPUT

8.26671687032.65745997035.0261805054

8.15871259852.67628172465.0552899346ObservationPredicted YResidualsStandard ResidualsPercentileY

8.41835560382.71164886845.078605420514158.58748209141251.78146592671.96115160083.84615384623493.6859307681

8.5126738222.75086958155.093750200824496.6316319442371.59238722330.582169511911.53846153853892.1495605542

8.60388049612.78042838385.115595729734818.5675545838-425.5972290292-0.666778275419.23076923084190.7993966818

8.65293067552.84808703235.14865659244718.2828539551-527.4834572734-0.826402255326.92307692314392.9703255546

8.80789315792.93792773635.176432178654716.3820496521-824.2324890979-1.291315544634.61538461544529.4491189078

8.8371183943.03050691765.19240114164580.9972951344-1087.3113643663-1.703478187642.30769230774868.2240191674

9.08878638293.17026885995.214663981674670.1975931414-140.7484742336-0.220509013504977.4541379019

84900.026380199277.42775770270.121305175957.69230769235410.3689480181

94960.5852423141492.19283159950.771112838765.38461538465452.7780739136

105320.631376125406.27449048810.636505563573.07692307695726.9058666131

116062.6180685223624.19828150650.977924256280.76923076926686.8163500288

127084.7668306752-199.643017605-0.312778415688.46153846156885.1238130702

138873.8836064896-18.4511828413-0.028907255696.15384615388855.4324236483

X Variable 1

Residuals


X Variable 2

Residuals


Y

Predicted Y

X Variable 1

Y


Y

Predicted Y

X Variable 2

Y


Sample Percentile

Y


Std Residual

Fitteed Severity

Standardized Residual

Fitted vs Standardized Residual

(1)(2)(3)(4)(1)(2)(3)(4)(5)(2)

WCHealthLag

YearSeverityIns IndexCPIResidualResidual(e(t)-e(t-1))^2e(t)^2FittedStd ResidualSUMMARY OUTPUT

19915,41011.7136.21,251.81,566,956.84,158.62.0

19924,86812.7140.3371.61,251.8774,732.8138,080.94,496.60.6Regression Statistics

19934,39313.6144.5(425.6)371.6635,511.3181,133.04,818.6(0.7)Multiple R0.8988759252

19944,19113.8148.3(527.5)(425.6)10,380.8278,238.84,718.3(0.8)R Square0.8079779288

19953,89214.3152.4(824.2)(527.5)88,060.0679,359.24,716.4(1.3)Adjusted R Square0.7695735146

19963,49414.5156.9(1,087.3)(824.2)69,210.51,182,246.04,581.0(1.7)Standard Error699.2105512592

19974,52915.1160.6(140.7)(1,087.3)895,981.319,810.14,670.2(0.2)Observations13

19984,97715.7163.077.4(140.7)47,600.95,995.14,900.00.1

19995,45316.1166.6492.277.4172,030.1242,253.84,960.60.8ANOVA

20005,72717.3172.2406.3492.27,382.0165,059.05,320.60.6dfSSMSFSignificance F

20016,68718.9177.1624.2406.347,490.8389,623.56,062.61.0Regression2.0020,571,421.1010285710.550894121.03867341820.0002610693

20026,88520.7179.9(199.6)624.2678,714.539,857.37,084.8(0.3)Residual10.004,888,953.95488895.394992245

20038,85523.8184.0(18.5)(199.6)32,830.5340.48,873.9(0.0)Total12.0025,460,375.05

Sum3,459,925.34,888,953.9

DW =0.7077025815CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Intercept5,903.563,623.591.60.1-2170.313977.4-2170.313977.4

X Variable 1667.63172.153.90.0284.11051.2284.11051.2

Ln(severity)Ln(HI)Ln(CPI)X Variable 2-70.2837.96-1.90.1-154.914.3-154.914.3

8.59607256692.46167134244.9141243937

8.49048447182.53863238354.9437829871

8.3877608892.60909084724.9732795076

8.34064678152.62693076534.9989000379RESIDUAL OUTPUTPROBABILITY OUTPUT

8.26671687032.65745997035.0261805054

8.15871259852.67628172465.0552899346ObservationPredicted YResidualsStandard ResidualsPercentileY

8.41835560382.71164886845.078605420514158.58748209141251.78146592671.96115160083.84615384623493.6859307681

8.5126738222.75086958155.093750200824496.6316319442371.59238722330.582169511911.53846153853892.1495605542

8.60388049612.78042838385.115595729734818.5675545838-425.5972290292-0.666778275419.23076923084190.7993966818

8.65293067552.84808703235.14865659244718.2828539551-527.4834572734-0.826402255326.92307692314392.9703255546

8.80789315792.93792773635.176432178654716.3820496521-824.2324890979-1.291315544634.61538461544529.4491189078

8.8371183943.03050691765.19240114164580.9972951344-1087.3113643663-1.703478187642.30769230774868.2240191674

9.08878638293.17026885995.214663981674670.1975931414-140.7484742336-0.220509013504977.4541379019

84900.026380199277.42775770270.121305175957.69230769235410.3689480181

94960.5852423141492.19283159950.771112838765.38461538465452.7780739136

105320.631376125406.27449048810.636505563573.07692307695726.9058666131

116062.6180685223624.19828150650.977924256280.76923076926686.8163500288

127084.7668306752-199.643017605-0.312778415688.46153846156885.1238130702

138873.8836064896-18.4511828413-0.028907255696.15384615388855.4324236483

X Variable 1

Residuals


X Variable 2

Residuals


Y

Predicted Y

X Variable 1

Y


Y

Predicted Y

X Variable 2

Y


Sample Percentile

Y


Std Residual

Fitteed Severity



(1)(2)(3)(4)(5)(6)

WCHealthLaw

YearSeverityIns IndexCPIReformFittedStd ResidualSUMMARY OUTPUT

19915,41011.7136.26.94,158.62.0

19924,86812.7140.37.84,496.60.6Regression Statistics

19934,39313.6144.574,818.6(0.7)Multiple R0.8988759252

19944,19113.8148.36.14,718.3(0.8)R Square0.8079779288

19953,89214.3152.45.64,716.4(1.3)Adjusted R Square0.7695735146

19963,49414.5156.95.34,581.0(1.7)Standard Error699.2105512592

19974,52915.1160.654,670.2(0.2)Observations13

19984,97715.7163.04.54,900.00.1

19995,45316.1166.64.34,960.60.8ANOVA

20005,72717.3172.245,320.60.6dfSSMSFSignificance F

20016,68718.9177.14.56,062.61.0Regression2.0020,571,421.1010285710.550894121.03867341820.0002610693

20026,88520.7179.95.87,084.8(0.3)Residual10.004,888,953.95488895.394992245

20038,85523.8184.06.38,873.9(0.0)Total12.0025,460,375.05

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

Median days away from workIntercept5,903.563,623.591.60.1-2170.313977.4-2170.313977.4

X Variable 1667.63172.153.90.0284.11051.2284.11051.2

YearAnnualLn(HI)Ln(CPI)X Variable 2-70.2837.96-1.90.1-154.914.3-154.914.3

199262.46167134244.9141243937

199362.53863238354.9437829871

199452.60909084724.9732795076

199552.62693076534.9989000379RESIDUAL OUTPUTPROBABILITY OUTPUT

199652.65745997035.0261805054

199752.67628172465.0552899346ObservationPredicted YResidualsStandard ResidualsPercentileY

199852.71164886845.078605420514158.58748209141251.78146592671.96115160083.84615384623493.6859307681

199962.75086958155.093750200824496.6316319442371.59238722330.582169511911.53846153853892.1495605542

200062.78042838385.115595729734818.5675545838-425.5972290292-0.666778275419.23076923084190.7993966818

200162.84808703235.14865659244718.2828539551-527.4834572734-0.826402255326.92307692314392.9703255546

8.80789315792.93792773635.176432178654716.3820496521-824.2324890979-1.291315544634.61538461544529.4491189078

8.8371183943.03050691765.19240114164580.9972951344-1087.3113643663-1.703478187642.30769230774868.2240191674

9.08878638293.17026885995.214663981674670.1975931414-140.7484742336-0.220509013504977.4541379019

84900.026380199277.42775770270.121305175957.69230769235410.3689480181

94960.5852423141492.19283159950.771112838765.38461538465452.7780739136

105320.631376125406.27449048810.636505563573.07692307695726.9058666131

116062.6180685223624.19828150650.977924256280.76923076926686.8163500288

127084.7668306752-199.643017605-0.312778415688.46153846156885.1238130702

138873.8836064896-18.4511828413-0.028907255696.15384615388855.4324236483

1251.7814659267

371.5923872233

-425.5972290292

-527.4834572734

-824.2324890979

-1087.3113643663

-140.7484742336

77.4277577027

492.1928315995

406.2744904881

624.1982815065

-199.643017605

-18.4511828413

X Variable 1

Residuals


1251.7814659267

371.5923872233

-425.5972290292

-527.4834572734

-824.2324890979

-1087.3113643663

-140.7484742336

77.4277577027

492.1928315995

406.2744904881

624.1982815065

-199.643017605

-18.4511828413

X Variable 2

Residuals


5410.36894801814158.5874820914

4868.22401916744496.6316319442

4392.97032555464818.5675545838

4190.79939668184718.2828539551

3892.14956055424716.3820496521

3493.68593076814580.9972951344

4529.44911890784670.1975931414

4977.45413790194900.0263801992

5452.77807391364960.5852423141

5726.90586661315320.631376125

6686.81635002886062.6180685223

6885.12381307027084.7668306752

8855.43242364838873.8836064896

Y

Predicted Y

X Variable 1

Y


5410.36894801814158.5874820914

4868.22401916744496.6316319442

4392.97032555464818.5675545838

4190.79939668184718.2828539551

3892.14956055424716.3820496521

3493.68593076814580.9972951344

4529.44911890784670.1975931414

4977.45413790194900.0263801992

5452.77807391364960.5852423141

5726.90586661315320.631376125

6686.81635002886062.6180685223

6885.12381307027084.7668306752

8855.43242364838873.8836064896

Y

Predicted Y

X Variable 2

Y


3493.6859307681

3892.1495605542

4190.7993966818

4392.9703255546

4529.4491189078

4868.2240191674

4977.4541379019

5410.3689480181

5452.7780739136

5726.9058666131

6686.8163500288

6885.1238130702

8855.4324236483

Sample Percentile

Y


1.9611516008

0.5821695119

-0.6667782754

-0.8264022553

-1.2913155446

-1.7034781876

-0.220509013

0.1213051759

0.7711128387

0.6365055635

0.9779242562

-0.3127784156

-0.0289072556

Std Residual

Fitteed Severity



MBD001F6C34.unknown

MBD001F6C36.unknown

MBD001F6C37.unknown

MBD001F6C35.unknown

MBD00129396.unknown

MBD0012EFC8.unknown

MBD001F6C33.unknown

MBD0012AE08.unknown

MBD00126489.unknown

MBD001271B2.unknown

Non-Linear RelationshipsThe model fit was of the form:Severity = a + b*YearA more common trend model is:SeverityYear=SeverityYear0*(1+t)(Year-Year0)T is the trend rateThis is an exponential trend modelCannot fit it with a line

Transformation of VariablesSeverityYear=SeverityYear0*(1+t)(Year-Year0)Log both sidesln(SevYear)=ln(SevYear0)+(Year-Year0)*ln(1+t) Y = a + x * bA line can be fit to transformed variables where dependent variable is log(Y)

Exponential Trend Cont.R2 declines and Residuals indicate poor fit

Chart1

0.319931207

0.1685457397

0.0200247847

-0.072886695

-0.1926139784

-0.3464156224

-0.1325699893

-0.0840491433

-0.0386398414

-0.0353870342

0.0737780761

0.0572059399

0.2630765566

Residuals

Predicted Y

Residual

Plot of Residuals

wORKERS cOMP




19915,846,56813,242,81217,484,55619,966,00621,578,47522,589,82123,263,62723,725,31324,095,56024,420,636199325,73224,58723,57422,27321,93321,62421,30021,07220,97920,941199313,0078,1005,8063,6882,9412,1451,6351,214878594199112,72516,48717,76818,58518,99219,47919,66519,85820,10120,3471991459.4552803.2275984.04751,074.30761,136.18761,159.70131,182.99651,194.74841,198.72441,200.20821991

199218,926,62519,336,89319,742,938199419,59219,55719,47819941,332979665199218,26018,57818,81319921,036.50741,258,629981,7181992

199315,809,90916,297,77016,794,672199518,65418,56318,46519951,6281,248827199317,02617,31517,6381993928.5745941.25159730011993

199413,775,47914,393,29814,894,261199618,89718,83818,90519961,7771,242919199417,12017,59617,9861994804.6425817.9869828.10301994

199511,859,70412,728,34013,420,655199719,86819,94920,21219972,2101,5531105199517,65818,39619,1071995671.6335691.9080702.39471995

199610,736,30512,212,29213,251,300199820,57420,92221,32719982,7921,9051413199617,78219,01719,9141996603.7738642.1776665.42631996(10,718,523)

19978,908,45411,546,92713,228,734199919,67920,48521,32319993,8452,1821633199715,83418,30319,6901997562.6155630.8762671.85041997(8,892,620)(11,528,624)

19984,560,5829,469,67112,495,703200020,57321,75122,68020008,6533,9452,487199811,92017,80620,1931998382.5992531.8247618.81361998(4,548,662)(9,451,865)(12,475,510)

19994,096,8359,576,909200123,05523,889200110,3565,479199912,69918,4101999322.6108520.20151999(4,084,136)(9,558,499)

20004,534,0352002200220000.020000.02000(4,534,035)

246195205056278

$19,009$13,269$7,461132947(13,166,833)(27,902,984)(34,722,657)






19912.2651.3201.1421.0811.0471.0301.0201.0161.01319911.2961.0781.0461.0221.0261.0101.0101.0121.012

19921.0221.02119921.0171.013

19931.0311.03019931.0171.019

19941.0451.035Claims19941.0281.022

19951.0731.054Alt19951.0421.039


19971.2961.1461.001199756182514,232,0741,193,6875,425,7615,623,86921,3233.795.0419,6903.501.081.0096.6%1.045.53051175051.05013,228,734671.850419971.1561.076

19982.0761.3201.01199856420994,186,6991,199,4695,386,1685,698,52022,6803.985.4220,1933.541.121.1395.5%1.065.52995949341.10012,495,703618.813619981.4941.134

19992.3380.01.05199954304703,708,3031,162,3714,870,6745,701,99423,8894.196.4418,4103.231.301.2089.7%1.15.645667161.2509,576,909520.201519991.450

20001.15200045863382,264,806935,4073,200,2135,274,2890.00.00.00.00.00.01.6769.8%1.1501.7004,534,0350.00002000

Bests DevRemainder



3.06694701651997192,490(13,209,044)(68,621.98)1.29760999461.12844975881.71.251.11.05Cumulative2.0791.4641.2961.2221.1751.1411.1191.0971.0811.070

1998255,931(12,475,510)(48,745.60)00IBNR lDF1.151.051.031.0111

2.09166666671.27333333331.131999559,796(9,558,499)(17,074.97)Severity1.80811.39461.25811.22251.17551.14121.11881.09691.08071.0700

20001,386,125(4,534,035)(3,271.01)

4.53740106542.01662269581.55124822751.36074405921.2599482031.19995066951.165000651.1365861.11431.11.97678837531.43621137281.2836050006

WC


Policy Year

Prior

19913944772913017,291.00$5,410.37

19923251066783416,678.00$4,868.22

19933049669422816,942.00$4,392.974375.3228120516

19942778566303316,630.00$4,190.804170.0435239382

19952612867134416,713.00$3,892.153866.8047950274

19962351667315716,731.00$3,493.693464.3488509134

19972853162997416,299.00$4,529.454476.8554840734

19983204764321151.0016,438.43$4,977.454894.9137009317

19993467863471831.0026,359.69$5,452.785310.5666156202

20003677063573391.016,420.57$5,726.91365491.3381123058

20013949157906161.025,905.80$6,686.82246164.6893537309

20023882953715211.055,639.55$6,885.12246678.1688919716

200342232414711671.154,769.05$8,855.4312

197015.71.1571

197110.91.1091.109

197211.11.1111.232099

19739.61.0961.350380504

197415.71.1571.5623902431

197514.21.1421.7842496577

197620.41.2042.1482365878

197716.11.1612.4941026785

197810.11.1012.746007049

197916.51.1653.1990982121

198015.21.1523.6853611403

198115.51.1554.256592117

198213.21.1324.8184622765

198381.085.2039392586

19848.21.0825.6306622778

1985101.16.1937285056

198651.056.5034149309

198710.91.1097.2122871583

1988151.158.2941302321

198912.91.1299.364073032

199012.91.12910.5720384531

199110.91.10911.7243906445

199281.0812.6623418961

19937.31.07313.5866928545

19941.81.01813.8312533259

19953.11.03114.260022179

19961.91.01914.5309626004

19973.61.03615.054077254

199841.0415.6562403442

199931.0316.1259275545

200071.0717.2547424833

20019.41.09418.8766882768

20029.71.09720.7077270396

2003151.1523.8138860955

Line Fit

WC

SUMMARY OUTPUT


(1)(2)(3)(4)(5)(6)(7)(8)




19923251066783416,678.00$4,868$3,9852$(467)-52,336.6325Standard Error1052.73

19933049669422816,942.00$4,393$4,2553$(943)-43,770.3216Observations13.00

19942778566303316,630.00$4,191$4,5254$(1,145)-33,434.259

19952612867134416,713.00$3,892$4,7965$(1,443)-22,886.804ANOVA


19972853162997416,299.00$4,529$5,3367$(806)00.00Regression113269748.7013269748.7011.970.01

19983204764321151.0016,438.43$4,977$5,6068$(358)1(358.10)1Residual1112190626.361108238.76

19993467863471831.0026,359.69$5,453$5,8769$1172234.454Total1225460375.05

20003677063573391.016,420.57$5,727$6,14610$39131,174.079


20023882953715211.055,639.55$6,885$6,68612$1,55057,747.8725Intercept3445.41619.375.560.002082.184808.642082.184808.64

200342232414711671.154,769.05$8,855$6,95613$3,520621,119.2936X Variable 1270.0278.033.460.0198.27441.7798.27441.77

Sum/Average$5,336749,143.61182.00

B270.02

3,445.41



13715.43164900311694.9372990153.84615384623493.6859307681

23985.4514762824882.77254288511.53846153853892.1495605542

34255.4713035617137.499021992919.23076923084190.7993966818

44525.4911308411-334.691734159326.92307692314392.9703255546

54795.5109581204-903.361397566234.61538461544529.4491189078

65065.5307853997-1571.844854631642.30769230774868.2240191674

75335.5506126791-806.1014937713504977.4541379019

85605.5704399584-628.116302056557.69230769235410.3689480181

95875.5902672377-422.812193324165.38461538465452.7780739136

106145.6100945171-418.70422790473.07692307695726.9058666131

116415.6299217964271.186428232480.76923076926686.8163500288

126685.6497490757199.474063994588.46153846156885.1238130702

136955.66957635511899.762847293396.15384615388855.4324236483

Line Fit

0

0

0

0

0

0

0

0

0

0

0

0

0

X Variable 1

Residuals

Expo Fit

0

0

0

0

0

0

0

0

0

0

0

0

0

Sample Percentile

Y


Sheet2

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

0000000

Y

Fitted Y

Year

Severity


Sheet3

WC

SUMMARY OUTPUT


(1)(2)(3)(4)(5)(5)(6)(7)(8)Multiple R0.6992898719

AccidentSeverityln(Severity)R Square0.489006325

YearYYFitted SevXAdjusted R Square0.4425523545

YearSeverityFittedStandard Error0.1904278372

19913944772913017,291.00$5,4108.60$3,9321$75-6(448.91)361991$5,410$3,932Observations13

19923251066783416,678.00$4,8688.49$4,1162$(467)-52,336.63251992$4,868$4,116

19933049669422816,942.00$4,3938.39$4,3093$(943)-43,770.32161993$4,393$4,309ANOVA

19942778566303316,630.00$4,1918.34$4,5114$(1,145)-33,434.2591994$4,191$4,511dfSSMSFSignificance F

19952612867134416,713.00$3,8928.27$4,7225$(1,443)-22,886.8041995$3,892$4,722Regression10.38172667260.381726672610.52668523720.0078114247

19962351667315716,731.00$3,4948.16$4,9436$(1,842)-11,841.8611996$3,494$4,943Residual110.39889037280.0362627612

19972853162997416,299.00$4,5298.42$5,1757$(806)00.001997$4,529$5,175Total120.7806170454

19983204764321151.0016,438.43$4,9778.51$5,4178$(358)1(358.10)11998$4,977$5,417

19993467863471831.0026,359.69$5,4538.60$5,6719$1172234.4541999$5,453$5,671CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

20003677063573391.016,420.57$5,7278.65$5,93710$39131,174.0792000$5,727$5,937Intercept8.23034398770.112037914373.460346314907.98375007628.47693789937.98375007628.4769378993

20013949157906161.025,905.80$6,6878.81$6,21511$1,35145,405.06162001$6,687$6,215X Variable 10.04579737220.01411545043.24448535780.00781142470.01472945960.07686528480.01472945960.0768652848

20023882953715211.055,639.55$6,8858.84$6,50712$1,55057,747.87252002$6,885$6,507

200342232414711671.154,769.05$8,8559.09$6,81113$3,520621,119.29362003$8,855$6,811

Sum/Average$5,3368.58$5,175749,143.61182.00

B270.02RESIDUAL OUTPUTPROBABILITY OUTPUT

3,445.41

ObservationPredicted YResidualsStandard ResidualsPercentileY

18.27614135990.3199312071.75477100673.84615384628.1587125985

28.32193873210.16854573970.924446164611.53846153858.2667168703

38.36773610430.02002478470.109832710419.23076923088.3406467815

48.4135334765-0.072886695-0.39977175226.92307692318.387760889

58.4593308487-0.1926139784-1.056456567734.61538461548.4183556038

68.5051282209-0.3464156224-1.900033748242.30769230778.4904844718

78.5509255931-0.1325699893-0.7271249834508.512673822

88.5967229653-0.0840491433-0.460995978357.69230769238.5960725669

98.6425203375-0.0386398414-0.211933290265.38461538468.6038804961

108.6883177097-0.0353870342-0.194092168173.07692307698.6529306755

118.73411508190.07377807610.404660833280.76923076928.8078931579

128.77991245410.05720593990.313765342788.46153846158.837118394

138.82570982630.26307655661.442932430296.15384615389.0887863829

Sheet3

0.0200247847

-0.072886695

-0.1926139784

-0.3464156224

-0.1325699893

-0.0840491433

-0.0386398414

-0.0353870342

0.0737780761

0.0572059399

0.2630765566

12

13

X Variable 1

Residuals

19.2307692308

26.9230769231

34.6153846154

42.3076923077

50

57.6923076923

65.3846153846

73.0769230769

80.7692307692

88.4615384615

96.1538461538

Sample Percentile

Y


199119911991199119915410.36894801818.3677361043

199219921992199219924868.22401916748.4135334765

199319931993199319934392.97032555468.4593308487

199419941994199419944190.79939668188.5051282209

199519951995199519953892.14956055428.5509255931

199619961996199619963493.68593076818.5967229653

199719971997199719974529.44911890788.6425203375

199819981998199819984977.45413790198.6883177097

199919991999199919995452.77807391368.7341150819

200020002000200020005726.90586661318.7799124541

200120012001200120016686.81635002888.8257098263

200220022002200220026885.12381307020

200320032003200320038855.43242364830

Y

Fitted Sev

Year

Severity


0.319931207

0.1685457397

0.0200247847

-0.072886695

-0.1926139784

-0.3464156224

-0.1325699893

-0.0840491433

-0.0386398414

-0.0353870342

0.0737780761

0.0572059399

0.2630765566

X Variable 1

Residuals


8.59607256698.2761413599

8.49048447188.3219387321

8.3877608898.3677361043

8.34064678158.4135334765

8.26671687038.4593308487

8.15871259858.5051282209

8.41835560388.5509255931

8.5126738228.5967229653

8.60388049618.6425203375

8.65293067558.6883177097

8.80789315798.7341150819

8.8371183948.7799124541

9.08878638298.8257098263

Y

Predicted Y

X Variable 1

Y


8.1587125985

8.2667168703

8.3406467815

8.387760889

8.4183556038

8.4904844718

8.512673822

8.5960725669

8.6038804961

8.6529306755

8.8078931579

8.837118394

9.0887863829

Sample Percentile

Y


Residuals

Predicted Y

Residual

Plot of Residuals

Severity

Fitted

Year

Severity

Exponential Trend

WCHealthSUMMARY OUTPUT

YearSeverityIns IndexCPIFittedResidualSUMMARY OUTPUT

19915,41011.7243906445136.24158.58748209141251.7814659267Regression Statistics

19924,86812.6623418961140.34496.6316319442371.5923872233Regression StatisticsMultiple R0.8229062628

19934,39313.5866928545144.54818.5675545838-425.5972290292Multiple R0.8988759252R Square0.6771747174

19944,19113.8312533259148.254718.2828539551-527.4834572734R Square0.8079779288Adjusted R Square0.6126096608

19953,89214.260022179152.354716.3820496521-824.2324890979Adjusted R Square0.7695735146Standard Error0.1587459978

19963,49414.5309626004156.854580.9972951344-1087.3113643663Standard Error699.2105512592Observations13

19974,52915.054077254160.554670.1975931414-140.7484742336Observations13

19984,97715.65624034421634900.026380199277.4277577027ANOVA

19995,45316

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Introduction to Generalized Linear Models Prepared by Louise Francis Francis Analytics and Actuarial...

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